For ease of discussion, the follow abbreviations are used herein: EKF for extended Kalman filter, FIS for fixed-interval smoother, FES for fixed-epoch smoother, SP for sequential processing, LS for least squares, IOD for initial orbit determination, RMS for root mean square, and VLS for variable-lag smoother.
Nonlinear multidimensional estimation problems require the use of nonlinear multidimensional estimation systems. Nonlinear estimation systems process measurements to estimate dynamical system states, and dynamical system states are driven by force models. All measurements have measurement errors, and all force models have force model errors. Thus all state estimates have estimation errors. One object of an estimation system is to calculate optimal state estimates—that is, state estimates with smallest estimation errors. Given no a priori state estimate, then a sequence of estimators must be applied to calculate an optimal multidimensional state estimate.
First an initial state estimate must be calculated using a nonlinear iterative technique. Every initial state estimate has very large estimation errors because the estimation problem is multidimensional and nonlinear, and because there does not exist an a priori state estimate to be refined since the state is completely unknown.
Given an initial state estimate with large estimation errors, the initial state estimate is input to the method of iterative least squares to generate a least squares state estimate, thereby producing a refined least squares state estimate with significantly reduced state estimate errors.
Given a refined least squares state estimate with sufficiently small state estimate errors, the least squares state estimate is used, together with an ad-hoc state estimate error covariance matrix, to initialize an optimal sequential real-time filter. The filter is then used to process measurements sequentially until the filter state estimate is fully initialized, after which the covariance matrix may be said to be realistic. Thereafter the filter is optimal in real time. Optimal predicted state estimates can be calculated from optimal filtered state estimates.
The user may trade the filter real time capability for state estimate accuracy improvement, by accepting state estimate time lag. For this purpose, during execution of the real-time filter, all filter calculations are saved for later use in a fixed-interval smoother (FIS). The user decides when to terminate the real-time filter run, and uses the filter state estimate and covariance to initialize an FIS. Using quantities calculated during the filter execution, the FIS is executed backwards in time to generate smoothed estimates with better accuracy than filtered estimates at the same times. The FIS also creates an FIS state estimate error covariance matrix with each FIS state estimate.
Spacecraft orbit determination provides an illustrative example of a nonlinear multi-dimensional orbit estimation problem with significant measurement errors and significant force modeling errors. The state estimate always contains six parameters for the orbit, one or more parameters for measurement biases, and various force model parameters. Specifically, the state estimate of a low Earth orbit (LEO) spacecraft comprises three components of spacecraft position, three components of spacecraft velocity, atmospheric density, ballistic coefficient, and measurement biases. By way of illustration and not as a limitation, other state parameters may include photon solar pressure, clock biases, spacecraft maneuver parameters, albedo acceleration, troposphere bias, ionosphere biases, station location biases, antenna phase center biases, and multi-path biases. The six components of spacecraft position and velocity are called the “orbit.” The general problem for orbit determination is first to estimate an initial orbit estimate, and thereafter to estimate corrections to previous state estimates using new measurements.
Currently estimators used for orbit determination involve an initial orbit determination (IOD), batch least squares differential corrections (LS), and sequential processing (SP). Operationally, the order in which these methods are used defines a functionally dependent sequence: An orbit estimate output from IOD can be used as orbit estimate input to initialize LS, and an orbit estimate output from LS can be used as orbit estimate input to initialize SP.
IOD methods input tracking measurements with tracking platform locations, and output spacecraft orbit estimates. No a priori orbit estimate is required, and when used, it is assumed that no a priori orbit estimate exists. Associated output orbit estimation error magnitudes are very large. IOD methods are nonlinear and six-dimensional. Refined measurement residual editing is not possible during IOD calculations because no a priori orbit estimate exists.
LS methods input tracking measurements with tracking platform locations and an a priori orbit estimate, and output a refined orbit estimate. An a priori orbit estimate is required. Associated output error magnitudes are small when compared to IOD outputs. LS methods consist of a sequence of linear LS corrections where sequence convergence is defined as a function of tracking measurement residual RMS (Root Mean Square). Each linear LS correction is characterized by a minimization of the sum of squares of tracking measurement residuals. LS methods produce refined orbit estimates in a batch mode, together with error covariance matrices that are optimistic. That is, LS orbit element error variance estimates are typically too small, relative to truth, by at least an order of magnitude. All LS methods require an inversion of an n×n LS information matrix for each LS iteration, where n is state estimate size. This is problematic when the information matrix is ill-conditioned. The information matrix is frequently ill-conditioned for orbit determination applications of LS. Operationally, LS may be the only method used, or it may be used to initialize SP. The existence of an a priori orbit estimate enables operational measurement residual editing, but LS methods frequently require inspection and manual measurement editing by human intervention. LS algorithms therefore require elaborate software mechanisms for measurement editing.
SP methods input tracking measurements with tracking platform locations, input an a priori state estimate (inclusive of orbit estimate), and input an a priori state error covariance matrix. An a priori state estimate is required, and an a priori state error covariance matrix is required. SP methods output refined state estimates in a sequential mode. SP filter methods are forward-time recursive sequential machines consisting of a repeating pattern of filter time update of the state estimate and filter measurement update of the state estimate. The filter time update propagates the state estimate forward, and the filter measurement update incorporates the next measurement. The recursive pattern includes an important interval of filter initialization. No state-sized matrix inversions are performed during filter calculations. The fixed-interval smoother (FIS) is a backward-time recursive sequential machine consisting of a repeating pattern of state estimate refinement using filter outputs and backwards transition. Matrix inversions are required by the FIS algorithm. Time transitions for both filter and smoother are CPU-dominated by numerical orbit propagations and state matrix multiply calculations.
In summary, IOD methods produce crude state estimates from measurements alone, whereas LS and SP methods produce refined corrections to a priori state estimates.
Measurement data consist of two additive unknown components: signal and noise. The signal is the unknown time-varying measurement component of interest. The noise is a random unknown time-varying functional that partially obscures the signal. The unknown true state is a function of the signal but not the noise. Thus every state estimate derived from measurements has state estimate errors due to measurement noise.
As an example, consider the measurement of the right ascension and declination angles of a telescope boresight that tracks celestial objects and/or spacecraft. Measurement noise is an additive composition of random white noise and random serially-correlated time-varying biases. Range and Doppler spacecraft measurements are derived from electronic hardware that consists of transmitters, receivers, transceivers, transponders, and other components. For GPS measurements, transmitters are deployed on GPS NAVSTAR spacecraft and receivers are deployed on USER spacecraft. Every component of electronic hardware produces random thermal noise (white noise) due to circuit resistance phenomena related to dynamics of electrons in the circuit. Random serially-correlated time-varying biases derive from clock phenomenology used in transmitters and receivers to create the range and Doppler measurements.
The forces applied to the satellite are modeled. For example, gravity models, atmospheric density models, ballistic models, solar photon pressure models, spacecraft surface description models, and thrust models are used in numerical orbit propagations to estimate the state of a LEO spacecraft. Because force models are inexact, the force models have errors and the state estimates they produce have errors.
As previously noted, a fixed-interval smoother (FIS) multi-dimensional nonlinear estimator uses a set of filter outputs. These filter outputs are stored while running the filter for later use in the FIS. The last filter output is used to initialize the FIS is the first smoother input. The filter runs forward with time. The FIS runs backwards with time. This incurs a significant operational throughput delay.
In an embodiment of a nonlinear multi-dimensional fixed-epoch smoother (FES), each FES is initialized by the EKF when the EKF reaches the fixed epoch specified by the user. The EKF processes measurements, but the FES does not process measurements. Rather, each FES moves measurement information, derived from the forward moving EKF state estimate, backwards to a fixed epoch. Each FES state estimate lags real-time.
In an embodiment, an extended Kalman filter (EKF) is combined with a fixed epoch smoother (FES) to produce a new variable lag smoother (VLS). The fixed epoch lags EKF measurement time-tags with variable time lag. The combination of EKF and FES is referred to herein as a variable lag smoother (VLS).
Described herein are two embodiments, a Fraser embodiment (FES/F), and a Carlton-Rauch embodiment (FES/CR). The FES/F does not require a state-sized matrix inverse calculation. The FES/CR requires the calculation of a state-sized covariance matrix inverse for each FES/CR execution. However, this is not meant as a limitation. Other FESs may be used in combination with various filters to produce a variable lag smoother providing the functionality described herein.